Reflective surface

ABSTRACT

The invention relates to a reflective surface substantially perpendicular to a vector field described by the equation: and a computer program for forming the reflective surface. The reflective surface is capable of providing a substantially undistorted wide-angle field of view. It is particularly useful as a diver&#39;s side mirror of a vehicle to provide an enlarged, substantially undistorted field of view which may be used to reduce or eliminate blind spots.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention is directed to a curved reflective surface capable ofproviding a substantially undistorted wide-angle field of view and acomputer program for forming said reflective surface. The invention mayhave applications in the field of illumination optics and driver's sidemirrors.

2. Description of the Related Technology

The U.S. Department of Transportation reported that lane changes andmerger (LCM) crashes alone accounted for approximately 244,000 crashesin the United States in 1994, causing 225 deaths and many seriousinjuries. This figure represents approximately 4% of all vehicularcrashes and is largely attributed to the minimal view provided by sideview mirror's. For most passenger side mirrors, the optical axis of theviewer's eye reflects off the mirror at approximately a 90° angle. Aflat driver side view mirror, however, typically provides a viewingangle of approximately 15°. Therefore, to ensure driver safety, there isa need to develop wide-angle side view mirrors, particularly wide-angledriver's side view mirrors, capable of enlarging the reflected field ofview when the mirror is viewed from the typical driver's position.

It is known to use curved mirrors to enlarge a field of view. Forexample, curved mirrors have previously been incorporated in the rearview and side view mirrors of automobiles, as disclosed in U.S. patentpublication nos. 2003/0039039, 2003/0081334 and 2004/0114260 and U.S.Pat. Nos. 6,979,090, 6,069,755, 5,980,050, 5,166,833, 4,580,881 and4,331,382.

Other forms of curved mirrors include side view mirrors that are capableof being manipulated into a curved configuration, such as is disclosedin Korean patent publication KR 1004847. The reflected images of thesemirrors, however, are generally significantly distorted, depending uponthe curvature and shape of the mirror.

Distorted images are non-perspective projections. A perspectiveprojection, by contrast, is formed by tracing a line from the imageplane through a point, known as the focal point or center of projection,until it touches an object in the scene. Pinhole cameras, for example,utilize this method for forming perspective images.

Historically, it was only possible to construct mirrors in spherical orparabolic shapes for traditional applications, such as in telescopedesigns. In recent years though, it has become possible, throughcomputer controlled machining, to create parts of almost any givenmathematical shape. Consequently, it is now possible to make mirrorswith an exactly prescribed geometry, even if it is highly irregular inshape. Although the technology exists for customizing the geometry of areflective surface, as far as the applicant is aware, there currentlyexists no reflective surface capable of enlarging the field of view of adriver side view mirror and reflecting a substantially undistortedimage.

SUMMARY OF THE INVENTION

The invention relates to a novel curved reflective surface capable ofprojecting a substantially undistorted image.

In second aspect, the invention is directed to a method and computerprogram for making the reflective surface.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram illustrating the derivation for theformula of the equation of the vector field W(x,y,z), given atransformation T from the image plane to the object plane.

FIG. 2 is a schematic diagram depicting source(x,y,z).

FIG. 3 illustrates the process of computing points on the surface M_(A)using the slice method.

FIG. 4 is a flowchart of a software process for generating a reflectivesurface in accordance with the present invention.

FIG. 5 illustrates a driver's side view mirror having a deflection angleof ψ° and a passenger's side view mirror having a deflection angle θ°.

FIG. 6 is a curved reflective surface defined by Equation 4.

FIG. 7( a) shows an experimental setup of two cars wherein Car A has aside view mirror incorporating the reflective surface of the presentinvention.

FIG. 7( b) shows an image of the view that a driver of Car A would seein the driver's side view mirror according to the present invention.

FIG. 7( c) shows an image of Car B as viewed by a driver sitting in CarA that puts his head out the window and looks back at Car B.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention is directed to a novel contoured reflectivesurface capable of projecting a substantially undistorted wide-anglefield of view or controlling illumination intensity. In a specificembodiment, the invention may be constructed as a driver's side viewmirror of a vehicle such as an automobile, train, watercraft, aircraft,motorcycles, buses or any other vehicle capable of projecting asubstantially undistorted reflection. The projection of a substantiallyundistorted reflection is referenced here when the reflective surface isviewed from the perspective of a seated driver. In the context of thispatent application, “driver” refers to the operator of any vehicleincluding at least automobiles, trains, watercraft, aircraft,motorcycles, buses, etc.

The driver's side view mirror can be provided with a customizable wideviewing angle which is preferably an angle of at least 30°, morepreferably, an angle of at least 40°, and, most preferably, an angle ofat least 45°, for reducing or eliminating blind spots in the driver'srear field of view. The field of view is referenced here as the viewwhen the reflective surface is viewed from the perspective of a seateddriver.

For the purpose of this patent application, the term “substantiallyundistorted” is defined by an error quantity, I_(e), of less than about15%. Preferably I_(e) is less than about 10%, more preferably, less thanabout 5% and most preferably, less than about 3%. Ie, which iscalculated according to Equation 1, represents the error formed by theprojection from a domain A, within an object plane, to an object planevia a mirror M. Namely, given a mirror M it induces a transform from theimage plane to the object plane by tracing light rays backwards from theimage plane, off of the mirror and to the object surface.

$\begin{matrix}{I_{e} = {\frac{1}{{diameter}\left( {T(A)} \right)}\left( {\int_{A}{{\begin{matrix}{{T\left( {y,z} \right)} -} \\{T_{M}\left( {y,z} \right)}\end{matrix}}^{2}{y}{z}}} \right)^{1/2}}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

To obtain a substantially undistorted reflection of an object surface atransform function, T, maps an object surface S to an image plane I in aprescribed way, i.e. T:I→S. T(A) is the image of a domain A, whichvaries depending on the application, in the image plane over which thereflective surface M is a graph. T_(M) is the transformation inducedfrom the image plane to the object surface by a mirror M. Equation 1provides a means of comparing the actions of T and T_(M) and may beinterpreted as an average, computed by considering the distance betweenan image of a point in the image plane under the given transform and thetransform induced by the mirror.

The invention is directed to a novel reflective surface x=f(y,z), whichmay be described by a collection of points (x, y, z) in space such that,when viewed along the positive x axis, the induced projection maps apoint source(x,y,z) on an image plane I to a point T(source(x,y,z)) onan object plane S. Note that this is the opposite of the direction thatreal light travels, but is framed this way for mathematical simplicity.Based on this correspondence, a vector field, W(x,y,z) is then definedon some subset of three dimensional space via the construction given inFIG. 1.

To compute the vector function W(x,y,z), a point [x,y,z] is projectedalong a ray to a point denoted as source(x,y,z), located in the imageplane I, as is depicted in FIG. 2. The given correspondence function Tmay be used to then compute the desired point T_(M)(source(x,y,z)) of atarget, 2. Then unit vectors, from point p to source point source(x,y,z)and target T_(M)(source(x,y,z)) may then be computed and added to definethe vector W(x,y,z). This procedure defines a non-zero vector fieldW(x,y,z) on a subset of three dimensional space.

If a surface M exists which is perpendicular to vector field W at eachthe points of M, then M is an exact solution to the problem. The lengthof vector field W is irrelevant so long as the tangent planes to M areperpendicular to vector field W.

As in FIG. 1, a vector W(x,y,z) may be defined by adding the unit vectorfrom point p to point source, to the unit vector from point p to pointT(source(x,y,z)). If a vector field is found to exist that is normal toa surface, then by use of elementary calculus M may be found. A test forthis situation is that an exact solution exists if and only if(∇×W)·W=0.In this case elementary calculus may be applied to find a solution,i.e., any one of the components of vector field W may be integrated withrespect to the appropriate variable to obtain an equation for M.

Because not every vector field is normal to a surface, i.e. a surfacenormal to vector field W, when M does not exist, an approximate solutionsurface M_(A), may be determined using a novel heuristic computation,known as the “slice method”. Given a proposed normal vector to thesurface, points on the surface can be found by integrating along“slices” of the distribution determined by vector field W, as shown inFIG. 3. By integrating along the slices, it is possible to generate areflective surface, where the height of the surface above P is theinitial condition, and integrating along a straight line to Q to definethe height above Q, as shown in FIG. 3.

To determine the coordinates of a vector W at a point [x,y,z], first asource point, source, in the image plane is calculated, source(x,y,z)=[source₁(x,y,z),source₂(x,y,z),source₃(x,y,z)], wheresource(x,y,z) is defined as the point in the image plane intersected bythe ray containing [x,y,z] and the coordinates corresponding to the eyeof an observer, 3, to obtain a source, as in FIG. 2. The point on theobject surface, target(x,y,z) is defined to be T(source(x,y,z)), whichin the case of the application of a side view mirror is given explicitlyin Equation 2 as:

target(x,y,z)=kλsource₂(x,y,z)[−sin(ψ),cos(ψ),0]+kλsource₃(x,y,z)[0,0,1]+k[cos(ψ),sin(ψ),0]  Equation2

where k is the distance between the reflective surface and object plane,ψ is the angle of deflection of the mirror as indicated in FIG. 5 and λis a magnification factor. Increasing λ will increase the field of viewof the observer. The vector field W is then defined as:

$\begin{matrix}{{W\left( {x,y,z} \right)} = {\frac{\begin{matrix}{{{target}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}}{\begin{matrix}{{{target}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}} + \frac{\begin{matrix}{{{source}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}}{\begin{matrix}{{{source}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

To obtain equation 4 the slice method is applied to W to yield acollection of points in three dimensional space that lie on the surfaceM_(A). Any standard available mathematical algorithm or software programmay then be used to generate a surface fit to these data points, i.e.generate a formula of the type given in Equation 4.

A flow chart of a software program suitable for generating thereflective surface of the present invention is shown in FIG. 4. Uponinputting parameters representing the image surface, source points,object surface, correspondence between the image and object surface, andobservation point, the program computes an algebraic expression ofW(x,y,z). The program then performs the slice method to generate samplepoints located on the reflective surface M_(A). Minor changes in thesoftware may be used to produce a reflective surface to accommodatedrivers of different heights, different fields of view, differentdeflection angle ψ, etc.

By appropriate choice of parameters for determining T, such as imageplane coordinates, object plane coordinates and magnification factor,the resultant reflective surface is capable of reflecting wide fields ofview. The reflective surface may also be tilted and adjusted to reduceor enlarge the field of view. Additionally, the reflective surfaceadvantageously projects a substantially undistorted image, as shown inFIG. 7( b).

The reflective surface of the present invention has numerousapplications. In one embodiment, it may be constructed as a driver sidemirror for a vehicle, such as an automobile, train, watercraft,aircraft, motorcycle or other vehicle to provide the driver or pilotwith an undistorted and enlarged field of view for eliminating blindspots. The invention is particularly effective as a driver's side viewmirror. Such a side view mirror is robust and is not substantiallydependent on the location of the driver's head.

The reflective surface may also have numerous applications in the fieldsof illumination optics or non-imaging optics, which are concerned withthe redistribution of radiation, at a prescribed intensity, from acollection of sources onto a target. J. H. McDermit and T. E. Horton,“Reflective optics for obtaining prescribed irradiative distributionsfrom collimated sources,” Applied Optics, 13:1444-1450, 1974; and R.Winston, J. Minano, and P. Benitez, “Nonimaging Optics,” ElsevierAcademic Press, 2005.

The reflective surface may be used to shape laser beams as analternative to, for example, D. L. Shealy, “Theory of geometricalmethods for design of laser beam shaping systems,” In Laser BeamShaping, Fred M. Dickey and Scott C. Holswade, eds., Proc. SPIE 4095,pages 1-15. SPIE Optical Engineering Press, San Diego, 2000., to enhancesolar collector designs (see e.g. R. Winston, “Selected Papers onNonimaging Optics” SPIE Optical Engineering Press, Bellingham, Wash.,1995), and control, focus or diffuse the illumination of any source,such as a light emitting diode, by directing the projection of thereflected light. Others have done some similar work on the illuminationproblem. See e.g. T. Glimm and V. Oliker, “Optical design of singlereflector systems and the monge-kantorovich mass transfer problem” J. ofMath, Sciences, 117:4096-4108, 2003; T. Glimm and V. Oliker, “Opticaldesign of two-reflector systems, the monge-kantorovich mass transferproblem and fermat's principle,” Indiana Univ Math. J, 53:1255-1277,2004; V. Oliker and S. Kochengin, “Computational algorithms forconstructing reflectors,” Computing and Visualization in Science,6:15-21, 2003; and H. Ries and J. Muschaweck, “Tailored freeform opticalsurfaces,” J. Opt. Soc. Am. A, 19:590-595, 2002.

These non-imaging optical applications utilize a simple means to realizea perspective projection, similar to the operation of pinhole cameras.Pinhole cameras typically operate by imaging a two-parameter family ofrays through a pinhole. Given the principle of reversibility ingeometric optics, which provides that rays may be thought of as eitherentering or leaving an optical system, i.e. sources may be considered tolie on the image plane, the pinhole camera may therefore be consideredthe source of the ray bundle. Therefore the theory behind forming aprescribed image with a pinhole camera and a curved mirror is equivalentto controlling illumination with a single source. Thus, for non-imagingapplications such as controlling the illumination of an LED, the eye ofthe observer, 3, as shown in FIG. 2, corresponds to the LED and theobject surface is the surface to be illuminated.

Example 1

A reflective surface M_(A) is illustrated in the context of a coordinatesystem containing a driver's side view mirror having an angle ofdeflection ψ=65° as shown in FIG. 5. Assuming that the eye of the driveris located at the coordinate [10,0,0] in a xyz space, a yz image planeat the coordinate [9,0,0] and a mirror centered at the origin [0,0,0],it is possible to compute a reflective surface M_(A) capable ofproviding an undistorted wide angle field of view. A typical driver'sside mirror subtends approximately a 15° field of view. To determine thevector field W(x,y,z), a source point was first calculated from (x,y,z),i.e. the point in the image plane intersected by the ray containing[x,y,z] and the eye of the observer at [10,0,0], resulting insource(x,y,z)=[9, y/(10-x), z/(10-x)].

T was calculated based on k=10 and λ=4. Then applying the programdepicted in FIG. 4, W(x,y,z) is determined and the method of slices,according to Equations 2 and 3, was applied over a region x=0, −1≦y≦1,−0.6≦z≦0.6, generating a set of data points on the reflective surface;these values may vary to suit different situations. A surface was fit tothese data points, and choosing units now to be millimeters and y=−60 mmto 60 mm and z=−36 mm to 36 mm results in Equation 4:

x(y,x)=−0.00166346z ²+0.0000000141941z ⁴+0.637076y+0.00000290062yz²−7.03493×10⁻¹¹ yz ⁴−0.001670555y ²+0.000000020558y ² z ²+0.00000290799y³−0.000000000114081y ³z²+0.00000000612366y ⁴−4.19119×10⁻¹¹ y ⁵,  Equation 4

FIG. 6 is a graph of the resultant reflective surface defined byEquation 4.

It should be noted that changes in these parameters may be made toaccommodate similar situations. For example the vertical height of themirror may be required to be different in which case W stays the same,but different integration parameters are chosen. This surface isdesigned to be centered at eye level and robust to motion.

The reflective surface of Example 1 may be incorporated in the driver'sside mirror shown in FIG. 7( b). FIG. 7( a) shows Car B, 5, located inthe blind spot of Car A's, 4, driver's side view mirror. FIG. 7( b)depicts the image of Car B, 5, seen by the driver of Car B, 5, viewed ina driver's side view mirror of the present invention. FIG. 7( c) depictsthe image that a driver of Car A, 4, would see if the driver extendedhis head out the driver's side window of Car A, 4, and turned his headaround to look at Car B, 5. Note that the mirror provides the driverwith an entirely visible and substantially undistorted 45° view of CarB, 16. By comparison, the field of view provided by standard driver sideview mirrors is approximately 13° to about 15°.

Having described the preferred embodiments of the invention which areintended to be illustrative and not limiting, it is noted thatmodifications and variations can be made by persons skilled in the artin light of the above teachings. It is therefore to be understood thatchanges may be made in the particular embodiments of the inventiondisclosed which are within the scope and spirit of the invention asoutlined by the appended claims. Having thus described the inventionwith the details and particularity required by the patent laws, theintended scope of protection is set forth in the appended claims.

1. A reflective surface substantially perpendicular to a vector fielddefined by the following equation:${W\left( {x,y,z} \right)} = {\frac{\begin{matrix}{{{target}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}}{\begin{matrix}{{{target}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}} + \frac{\begin{matrix}{{{source}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}}{\begin{matrix}{{{source}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}}}$
 2. The reflective surface of claim 1, wherein thereflective surface has an image error quantity, I_(e), of less thanabout 15%, calculated according to the following equation:$I_{e} = {\frac{1}{{diameter}\left( {T(A)} \right)}\left( {\int_{A}{{{{T\left( {y,z} \right)} - {T_{M}\left( {y,z} \right)}}}^{2}{y}{z}}} \right)^{1/2}}$wherein A is the image of a domain in the image plane over which thesurface M is a graph and T_(M) is the transformation induced from theimage plane to the object surface by reflecting at least one ray off M.3. The reflective surface of claim 2, wherein the reflective surface hasan image error quantity of less than about 10%.
 4. The reflectivesurface of claim 2, wherein the reflective surface has an image errorquantity of less than about 5%.
 5. The reflective surface of claim 2,wherein the reflective surface has an image error quantity of less thanabout 3%.
 6. The reflective surface of claim 1, wherein the reflectivesurface is capable of reflecting a substantially undistorted image fromthe perspective of a seated driver.
 7. The reflective surface of claim2, wherein said reflective surface is capable of reflecting at least a30° field of view when viewed from the perspective of a seated driver.8. The reflective surface of claim 2, wherein said reflective surface iscapable of reflecting at least a 40° field of view when viewed from anperspective of a seated driver.
 9. The reflective surface of claim 2,wherein said reflective surface is capable of reflecting at least a 45°field of view when viewed from a perspective of a seated driver.
 10. Thereflective surface of claim 2, wherein said reflective surface isadjustable to enlarge or reduce the reflected field of view from thesame point of observation.
 11. A driver's side view mirror for a vehiclecomprising the reflective surface of claim
 1. 12. A driver's side viewmirror for a vehicle comprising the reflective surface of claim
 2. 13. Adriver's side view mirror for a vehicle comprising the reflectivesurface of claim
 7. 14. A computer program comprising one or morecomputer-readable storage media having stored thereoncomputer-executable instructions for implementing a method forgenerating a reflective surface comprising the steps of: inputting dataincluding at least an image surface, a domain of an image surface, anobject surface, a correspondence T and the coordinates of an eye of anobserver, computing W(x,y,z) as an algebraic expression from Equations2-3:target(x,y,z)=kλsource₂(x,y,z)[−sin(ψ),cos(ψ),0]+kλsource₃(x,y,z)[0,0,1]+k[cos(ψ),sin(ψ),0]  Equation 3 wherein k is the distancebetween the reflective surface and object plane, ψ is the angle ofdeflection of the mirror and X is a magnification factor;$\begin{matrix}{{W\left( {x,y,z} \right)} = {\frac{\begin{matrix}{{{target}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}}{\begin{matrix}{{{{target}\left( {x,y,z} \right)}} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}} + \frac{\begin{matrix}{{{source}\left( {x,y,z} \right)} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}}{\begin{matrix}{{{{source}\left( {x,y,z} \right)}} -} \\\left\lbrack {x,y,z} \right\rbrack\end{matrix}}}} & {{Equation}\mspace{14mu} 4}\end{matrix}$ and generating points on a reflective surface.
 15. Thecomputer program of claim 14, wherein the reflective surface reflects afield of view of at least 30° when viewed from a perspective of a seateddriver.
 16. The computer program of claim 14, wherein the reflectivesurface reflects a field of view of at least 40° when viewed from aperspective of a seated driver.
 17. The computer program of claim 14,wherein the reflective surface reflects a field of view of at least 45°when viewed from a perspective of a seated driver.
 18. The computerprogram of claim 14, wherein the reflective surface has an image errorquantity, I_(e), of less than about 15%, calculated according to:$I_{e} = {\frac{1}{{diameter}\left( {T(A)} \right)}\left( {\int_{A}{{{{T\left( {y,z} \right)} - {T_{M}\left( {y,z} \right)}}}^{2}{y}{z}}} \right)^{1/2}}$wherein V is the image of a domain in the image plane over which thereflective surface M is a graph and T_(M) is the transformation inducedfrom the image plane to the object surface by M.
 19. The reflectivesurface of claim 18, wherein the reflective surface has an image errorquantity of less than about 10%.
 20. The reflective surface of claim 18,wherein the reflective surface has an image error quantity of less thanabout 5%.